# Properties

 Label 5520.2.a.bx Level $5520$ Weight $2$ Character orbit 5520.a Self dual yes Analytic conductor $44.077$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.0774219157$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + ( - \beta_{2} + 2) q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + (-b2 + 2) * q^7 + q^9 $$q - q^{3} - q^{5} + ( - \beta_{2} + 2) q^{7} + q^{9} + (\beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} + 3 \beta_1 - 1) q^{13} + q^{15} + ( - 3 \beta_1 - 1) q^{17} + (\beta_{2} - \beta_1 + 1) q^{19} + (\beta_{2} - 2) q^{21} - q^{23} + q^{25} - q^{27} + (2 \beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 3 \beta_{2} + 2) q^{31} + ( - \beta_{2} - \beta_1 + 1) q^{33} + (\beta_{2} - 2) q^{35} + (3 \beta_{2} - 2 \beta_1 - 2) q^{37} + (\beta_{2} - 3 \beta_1 + 1) q^{39} + ( - 2 \beta_{2} + \beta_1 - 5) q^{41} + (4 \beta_{2} - 2 \beta_1 + 2) q^{43} - q^{45} + (3 \beta_{2} - 3 \beta_1 + 7) q^{47} + ( - 5 \beta_{2} + 2 \beta_1 + 1) q^{49} + (3 \beta_1 + 1) q^{51} + (3 \beta_1 - 7) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{55} + ( - \beta_{2} + \beta_1 - 1) q^{57} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{59} + (\beta_{2} - \beta_1 - 5) q^{61} + ( - \beta_{2} + 2) q^{63} + (\beta_{2} - 3 \beta_1 + 1) q^{65} + ( - \beta_{2} + 2 \beta_1 + 4) q^{67} + q^{69} + (6 \beta_{2} - 5 \beta_1 + 3) q^{71} + (3 \beta_{2} + 3 \beta_1 - 1) q^{73} - q^{75} + (3 \beta_{2} - \beta_1 - 7) q^{77} + q^{81} + ( - 4 \beta_{2} + \beta_1 + 1) q^{83} + (3 \beta_1 + 1) q^{85} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{87} + (2 \beta_1 - 4) q^{89} + ( - 5 \beta_{2} + 5 \beta_1 - 1) q^{91} + (3 \beta_{2} - 2) q^{93} + ( - \beta_{2} + \beta_1 - 1) q^{95} + (2 \beta_{2} - 4 \beta_1 + 2) q^{97} + (\beta_{2} + \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^3 - q^5 + (-b2 + 2) * q^7 + q^9 + (b2 + b1 - 1) * q^11 + (-b2 + 3*b1 - 1) * q^13 + q^15 + (-3*b1 - 1) * q^17 + (b2 - b1 + 1) * q^19 + (b2 - 2) * q^21 - q^23 + q^25 - q^27 + (2*b2 - 3*b1 - 1) * q^29 + (-3*b2 + 2) * q^31 + (-b2 - b1 + 1) * q^33 + (b2 - 2) * q^35 + (3*b2 - 2*b1 - 2) * q^37 + (b2 - 3*b1 + 1) * q^39 + (-2*b2 + b1 - 5) * q^41 + (4*b2 - 2*b1 + 2) * q^43 - q^45 + (3*b2 - 3*b1 + 7) * q^47 + (-5*b2 + 2*b1 + 1) * q^49 + (3*b1 + 1) * q^51 + (3*b1 - 7) * q^53 + (-b2 - b1 + 1) * q^55 + (-b2 + b1 - 1) * q^57 + (-2*b2 + 3*b1 - 5) * q^59 + (b2 - b1 - 5) * q^61 + (-b2 + 2) * q^63 + (b2 - 3*b1 + 1) * q^65 + (-b2 + 2*b1 + 4) * q^67 + q^69 + (6*b2 - 5*b1 + 3) * q^71 + (3*b2 + 3*b1 - 1) * q^73 - q^75 + (3*b2 - b1 - 7) * q^77 + q^81 + (-4*b2 + b1 + 1) * q^83 + (3*b1 + 1) * q^85 + (-2*b2 + 3*b1 + 1) * q^87 + (2*b1 - 4) * q^89 + (-5*b2 + 5*b1 - 1) * q^91 + (3*b2 - 2) * q^93 + (-b2 + b1 - 1) * q^95 + (2*b2 - 4*b1 + 2) * q^97 + (b2 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 + 6 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^15 - 6 * q^17 + 2 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 6 * q^35 - 8 * q^37 - 14 * q^41 + 4 * q^43 - 3 * q^45 + 18 * q^47 + 5 * q^49 + 6 * q^51 - 18 * q^53 + 2 * q^55 - 2 * q^57 - 12 * q^59 - 16 * q^61 + 6 * q^63 + 14 * q^67 + 3 * q^69 + 4 * q^71 - 3 * q^75 - 22 * q^77 + 3 * q^81 + 4 * q^83 + 6 * q^85 + 6 * q^87 - 10 * q^89 + 2 * q^91 - 6 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.34292 −1.81361 0.470683
0 −1.00000 0 −1.00000 0 −0.489289 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.71083 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 4.77846 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5520.2.a.bx 3
4.b odd 2 1 2760.2.a.r 3
12.b even 2 1 8280.2.a.bm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.r 3 4.b odd 2 1
5520.2.a.bx 3 1.a even 1 1 trivial
8280.2.a.bm 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}^{3} - 6T_{7}^{2} + 5T_{7} + 4$$ T7^3 - 6*T7^2 + 5*T7 + 4 $$T_{11}^{3} + 2T_{11}^{2} - 14T_{11} - 32$$ T11^3 + 2*T11^2 - 14*T11 - 32 $$T_{13}^{3} - 34T_{13} + 76$$ T13^3 - 34*T13 + 76 $$T_{17}^{3} + 6T_{17}^{2} - 27T_{17} - 86$$ T17^3 + 6*T17^2 - 27*T17 - 86 $$T_{19}^{3} - 2T_{19}^{2} - 6T_{19} + 8$$ T19^3 - 2*T19^2 - 6*T19 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 6 T^{2} + 5 T + 4$$
$11$ $$T^{3} + 2 T^{2} - 14 T - 32$$
$13$ $$T^{3} - 34T + 76$$
$17$ $$T^{3} + 6 T^{2} - 27 T - 86$$
$19$ $$T^{3} - 2 T^{2} - 6 T + 8$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 6 T^{2} - 31 T - 122$$
$31$ $$T^{3} - 6 T^{2} - 51 T + 64$$
$37$ $$T^{3} + 8 T^{2} - 35 T + 22$$
$41$ $$T^{3} + 14 T^{2} + 41 T - 58$$
$43$ $$T^{3} - 4 T^{2} - 92 T + 496$$
$47$ $$T^{3} - 18 T^{2} + 42 T + 272$$
$53$ $$T^{3} + 18 T^{2} + 69 T - 2$$
$59$ $$T^{3} + 12 T^{2} + 5 T - 64$$
$61$ $$T^{3} + 16 T^{2} + 78 T + 116$$
$67$ $$T^{3} - 14 T^{2} + 49 T - 4$$
$71$ $$T^{3} - 4 T^{2} - 235 T + 1376$$
$73$ $$T^{3} - 138T - 596$$
$79$ $$T^{3}$$
$83$ $$T^{3} - 4 T^{2} - 95 T - 164$$
$89$ $$T^{3} + 10 T^{2} + 16 T - 16$$
$97$ $$T^{3} - 2 T^{2} - 64 T - 128$$